{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<center> <h1> Basic probability </h1></center>\n",
    "# 1. Basic rules on probability\n",
    "* Product rule:\n",
    "$$ p(A,B)=p(A)p(B|A)$$\n",
    "* Sum rule:\n",
    "$$ p(A)=\\sum_b p(A,B)=\\sum_b p(A|B=b)p(B=b)$$\n",
    "* Conditional probability:\n",
    "$$ p(B|A)=\\frac{P(A,B)}{P(A)}$$\n",
    "* Bayes rule:\n",
    "$$p(B|A)=\\frac{p(A|B)p(B)}{p(A)}$$\n",
    "* Independent:\n",
    "$$ A \\perp B \\leftrightarrow p(A,B)=p(A)p(B)$$\n",
    "* Conditional independent:\n",
    "$$ A \\perp B |C \\leftrightarrow p(A,B|C)=p(A|C)p(B|C)$$\n",
    "\n",
    "# 2. Some common discrete distributions\n",
    "## The Bernoulli and binomial distributions\n",
    "The **Bernoulli distribution** is the simplest distribution for a binary random variable, such as  we toss a coin once and the result whether it turn out to be \"head\". Let $ X \\in \\lbrace0,1 \\rbrace$ be a binary random variable, the Bernoulli distribution is writen as $X \\sim Ber(\\theta)$, where $\\theta$ is the probability of $X=1$.\n",
    "$$ Ber(x|\\theta)=\\theta^{\\mathbb{1}(x=1)}(1-\\theta)^{\\mathbb{1}(x=0)}$$\n",
    "In other words\n",
    "$$ Ber(x|\\theta)=\n",
    "\\begin{cases}\n",
    "\\theta & x=1 \\\\\n",
    "1-\\theta & x=0\n",
    "\\end{cases}\n",
    "$$\n",
    "Suppose we repeat to sample from the $Ber(x|\\theta)$ n times, a random variable $Y \\in \\lbrace 0,1,\\ldots,n \\rbrace$ is defined as  the times when $x=1$.Then we say $Y$ has a **binomial dirstribution**, writen as $Y \\sim Bin(n,\\theta)$.\n",
    "$$ Bin(y=k\\,|\\,n,\\theta)=\\frac{n!}{(n-k)!k!} \\theta^k (1-\\theta)^{(1-k)}$$\n",
    "The binomial distribution has the following mean and variance:\n",
    "$$ mean=n\\theta, \\quad var=n\\theta(1-\\theta)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb891d7b9b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import binom\n",
    "import seaborn as sb\n",
    "import matplotlib.pyplot as plt\n",
    "rv=binom(n=100,p=0.2)\n",
    "data=rv.rvs(size=5000,random_state=42)\n",
    "ax=sb.distplot(data,\n",
    "              kde=True,\n",
    "              color='blue',\n",
    "              hist_kws={\"linewidth\":25,\"alpha\":1})\n",
    "ax.set(xlabel='Binomial', ylabel='Frequency')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The multinoulli and multinomial distributions\n",
    "The multinoulli distribution is the simplest distribution for a category random variable, such as the outcome of tossing a $K$-side die. Let $X \\in \\lbrace 1,2,\\ldots,K\\rbrace$ be a category random variable, the multinoulli distribution is writen as $X \\sim Cat(\\vec{\\theta})$,where $\\theta_i$ is the probability of $X=i$ and $\\sum_{i=1}^K \\theta_i=1$.\n",
    "$$ Cat(x|\\vec{\\theta})=\\prod_{i=1}^K \\theta_i^{\\mathbb{1}(x=i)}$$\n",
    "In other words\n",
    "$$\n",
    "Cat(x|\\vec{\\theta})= \n",
    "\\begin{cases}\n",
    "\\theta_1  & x=1 \\\\\n",
    "\\theta_2  & x=2  \\\\\n",
    "\\ldots           \\\\\n",
    "\\theta_K  & x=K\n",
    "\\end{cases}\n",
    "$$\n",
    "Suppose we repeat to sample from $Cat(x|\\vec{\\theta})$ n times, let $\\vec{y}=(n_1,n_2,\\ldots,n_K)$ be a random vector, where $n_i$ is the number of times $x=i$. Then $\\vec{y}$ has a multinomial distribution,writen as $\\vec{y} \\sim Mu(n,\\vec{\\theta})$\n",
    "$$Mu(\\vec{y}|n,\\vec{\\theta})=\\frac{n!}{y_1\\,!,y_2\\,!,\\ldots,y_K\\,!}\\prod_{i=1}^K \\theta_i^{y_i}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3.Some common continuous distributions\n",
    "## Gaussian (normal) distribution\n",
    "The most widely used distribution in statistics is the Gaussian or normal distribution.\n",
    "$$\\mathcal{N}(x|\\mu,\\sigma^2)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}$$\n",
    "There are several reasons for its widely used.\n",
    "* It has two parameter which are easy to interpret, and capture some of the most basic properties of a distribution, namely its mean $\\mu$ and its variance $\\sigma^2$.\n",
    "* The central limit theorem tells us that sums of independent random variables has an approximately Gaussian distribution, making it a good choice for modeling residual errors or \"noise\".\n",
    "* The Gaussian distribution has maximum entropy subject to constraint of having specified mean and variance.\n",
    "\n",
    "## Student t distribution\n",
    "One problem with the Gaussian distribution is that it is sensitive to outliers, since the log-probability decays quadratically with distance from the center. A more robust distribution is the **Student t distribution** as following\n",
    "$$\\mathcal{T}(x|\\mu,\\sigma^2,\\nu) \\propto \\left[1+\\frac{1}{\\nu} \\left(\\frac{x-\\mu}{\\sigma}\\right)^2\\right]^{-\\left(\\frac{\\nu+1}{2}\\right)}$$\n",
    "where $\\nu >0$ is called the **degree of freedom**. We note that the distribution has the following properties\n",
    "$$mean=\\mu,\\quad mode=\\mu,\\quad var=\\frac{\\nu\\sigma^2}{\\nu-2}$$\n",
    "It's common to use $\\nu=4$, which gives good performance in a range of problems. For $\\nu \\gg 5$, the student distribution rapidly approaches a Gaussian distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'With outliers')"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb88e6baf98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import norm\n",
    "import scipy.stats as ss\n",
    "import numpy as np\n",
    "import seaborn as sb\n",
    "import matplotlib.pyplot as plt\n",
    "norm_data=norm.rvs(size=2000,random_state=42)\n",
    "norm_data_outlier=np.concatenate((norm_data,np.ones(50)*10))\n",
    "plt.figure(figsize=(12,6))\n",
    "\n",
    "plt.subplot(1,2,1)\n",
    "ax=sb.distplot(norm_data,\n",
    "              kde=False,\n",
    "              color='k',\n",
    "              fit=norm,\n",
    "              fit_kws={\"color\":\"red\",\"label\":\"Gaussian distribution\"},\n",
    "              hist_kws={\"linewidth\":25,\"alpha\":1})\n",
    "sb.distplot(norm_data,\n",
    "              kde=False,\n",
    "              color='k',\n",
    "              fit=ss.t,\n",
    "              fit_kws={\"color\":\"blue\",\"label\":\"Student distribution\"},\n",
    "              hist_kws={\"linewidth\":25,\"alpha\":1})\n",
    "plt.legend()\n",
    "plt.title(\"No outliers\",fontsize=20)\n",
    "\n",
    "plt.subplot(1,2,2)\n",
    "ax=sb.distplot(norm_data_outlier,\n",
    "              kde=False,\n",
    "              color='k',\n",
    "              fit=norm,\n",
    "              fit_kws={\"color\":\"red\",\"label\":\"Gaussian distribution\"},\n",
    "              hist_kws={\"linewidth\":25,\"alpha\":1})\n",
    "sb.distplot(norm_data_outlier,\n",
    "              kde=False,\n",
    "              color='k',\n",
    "              fit=ss.t,\n",
    "              fit_kws={\"color\":\"blue\",\"label\":\"Student distribution\"},\n",
    "              hist_kws={\"linewidth\":25,\"alpha\":1})\n",
    "plt.legend()\n",
    "plt.title(\"With outliers\",fontsize=20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Laplace distribution\n",
    "Another distribution which is more robust than Gaussian distribution is the **Laplace distribution**.\n",
    "$$Lap(x|\\mu,b)=\\frac{1}{2b}exp\\left(-\\frac{|x-\\mu|}{b}\\right)$$\n",
    "This distribution has the following properties\n",
    "$$ mean=\\mu,\\quad mode=\\mu,\\quad var=2b^2$$\n",
    "The Laplace distribution is much higher than Gaussian at $\\mu$ and with heavy tails, which means \"high peak and heavy tails."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb88cc79240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sb\n",
    "from scipy.stats import norm\n",
    "from scipy.stats import laplace\n",
    "norm_data=norm.rvs(size=1000,random_state=42)\n",
    "laplace_data=laplace.rvs(scale=(np.sqrt(0.5)),size=1000,random_state=42)\n",
    "plt.figure(figsize=(8,8))\n",
    "ax=sb.distplot(norm_data,\n",
    "              hist=False,\n",
    "              kde=False,\n",
    "              fit=norm,\n",
    "              fit_kws={\"color\":\"r\",\"label\":\"Gaussian distribution\"})\n",
    "ax=sb.distplot(laplace_data,\n",
    "              hist=False,\n",
    "              kde=False,\n",
    "              fit=laplace,\n",
    "              fit_kws={\"color\":\"b\",\"label\":\"Laplace distribution\"})\n",
    "plt.legend(fontsize=15)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The gamma distribution\n",
    "The **gamma distribution** is a flexible distribution for positive real random variable.It's defined as following\n",
    "$$ Ga(T|shape=a,rate=b)=\\frac{b^a}{\\Gamma(a)}T^{a-1}e^{-Tb}$$\n",
    "The gamma distribution is of great importance in the estimation of the variance of the univariate Gaussian distribution.\n",
    "\n",
    "## The beta distribution\n",
    "The **beta distribution** has support over the interval $[0,1]$ and is defined as follows\n",
    "$$ Beta(\\theta|a,b)=\\frac{1}{B(a,b)}x^{a-1}(1-x)^{b-1}$$\n",
    "The beta distribution is of great importance in the estimation of the $\\theta$ of the binomial distributions.\n",
    "\n",
    "# 4. Joint probability distributions\n",
    "A **joint probability distribution** has the form $p(x_1,x_2,\\ldots,x_D)$ for a set of variables.\n",
    "## Covariance and correlation\n",
    "The covariance between two variable $X$ and $Y$ measures the degree to which $X$ and $Y$ are(linearly) related.\n",
    "$$\n",
    "\tcov[X,Y]=E[(X-E[X])(Y-E[Y])]\n",
    "$$\n",
    "For a d-dimensional random vector $\\vec{x}$, the covariance matrix is defined to be\n",
    "$$\n",
    "\tcov[\\vec{x}]=E\\left[(x-E[\\vec{x}])(x-E[\\vec{x}])^T\\right]\n",
    "$$\n",
    "We can normalize the covariance between $X$ and $Y$ as following\n",
    "$$\n",
    "\tcorr[X,Y]=\\frac{cov[X,Y]}{\\sqrt{var[X]var[Y]}}\n",
    "$$\n",
    "which called (Pearson) correlation coefficient between $X$ and $Y$.\n",
    "\n",
    "## The multivariate Gaussian\n",
    "The most widely used joint probability density function for continuous variables is multivariate Gaussian. \n",
    "$$\n",
    "\t\\mathcal{N}(\\vec{x}|\\vec{\\mu},\\Sigma)=\\frac{1}{(2\\pi)^{D/2}|\\Sigma^{1/2}}exp\\left[-\\frac{1}{2}(\\vec{x}-\\vec{\\mu})^T\\Sigma^{-1}(\\vec{x}-\\vec{\\mu})\\right]\n",
    "$$\n",
    "The $\\vec{\\mu}$ is the mean value and the $\\Sigma$ is the covariance matrix. We will discuss it more detailly in the future.\n",
    "\n",
    "## The Dirichlet distribution\n",
    "The **Dirichlet distribution** is a multivariate generalization of the **beta distribution**, which describle the joint probability distribution of a set of $\\theta_i$ which satisfy\n",
    "$$\n",
    "\t0\\leq \\theta_k\\leq1, \\sum_{k=1}^K \\theta_k=1\n",
    "$$\n",
    "The Dirichlet distribution is defined:\n",
    "$$\n",
    "\tDir(\\theta|\\alpha)=\\frac{1}{B(\\alpha)}\\prod_{k=1}^K \\theta_k^{\\alpha_k-1}\n",
    "$$\n",
    "\n",
    "## The Wishart distribution\n",
    "The **Wishart distribution** is a multivariate generalization of the **gamma distribution**. It is a family of probability distributions defined over symmetric, nonnegative-definite matrix-valued random variables (\"random matrices\"). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics.\n",
    "\\begin{align*}\n",
    "\\mathcal{W}(\\Lambda|W,\\nu) &=B(W,\\nu)|\\Lambda|^{(\\nu-D-1)/2}exp\\left(-\\frac{1}{2}Tr(W^{-1}\\Lambda) \\right) \\\\\n",
    "B(W,\\nu) &=|W|^{-\\nu/2}\\left(2^{\\nu D/2}\\pi^{D(D-1)/4} \\prod_{i=1}^D \\Gamma \\left(\\frac{\\nu+1-i}{2} \\right) \\right)^{-1}\n",
    "\\end{align*}\n",
    "\n",
    "where $\\nu \\geq D$ is called the **degrees of freedom** of the distribution, $W$ is a positive definite $D \\times D$ scale matrix. The mean and mode of the Wishart distribution is as following\n",
    "\\begin{align*}\n",
    "mean &=\\nu W  \\\\\n",
    "mode &=(\\nu-D-1) W\n",
    "\\end{align*}"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python [default]",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}
